D3-Branes Wrapped on a Spindle
Pietro Ferrero ,1 Jerome P. Gauntlett ,2 Juan Manuel P´erez Ipiña ,1 Dario Martelli ,3,4 and James Sparks1 1
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom 2Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2AZ, United Kingdom
3
Dipartimento di Matematica, Universit `a di Torino, Via Carlo Alberto 10, 10123 Torino, Italy 4INFN, Sezione di Torino & Arnold-Regge Center, Via Pietro Giuria 1, 10125 Torino, Italy
(Received 22 December 2020; accepted 19 February 2021; published 18 March 2021) We construct supersymmetric AdS3×Σ solutions of minimal gauged supergravity in D ¼ 5, where Σ is a two-dimensional orbifold known as a spindle. Remarkably, these uplift on S5, or more generally on any regular Sasaki-Einstein manifold, to smooth solutions of type IIB supergravity. The solutions are dual to d¼ 2, N ¼ ð0; 2Þ SCFTs and we show that the central charge for the gravity solution agrees with a field theory calculation associated with D3-branes wrapped onΣ.
DOI:10.1103/PhysRevLett.126.111601
Introduction.—Important insights into strongly coupled supersymmetric conformal field theories (SCFTs) can be obtained by realizing them as the renormalization group fixed points of compactifications of higher-dimensional field theories. Such SCFTs can be constructed by starting with the field theories arising on the world volumes of branes in string theory or M theory, wrapping them on a compact manifoldΣ, and then flowing to the infrared (IR). In favorable circumstances these configurations can be described within the AdS=CFT correspondence by brane solutions of D¼ 10=11 supergravity. Such solutions have a boundary of the form AdSdþ1× M, where M is a compact manifold, which describes the ultraviolet (UV) of the SCFT in d spacetime dimensions. They also have near horizon geometries of the schematic form AdSdþ1−n×Σ × M, which describes the SCFT in d− n dimensions arising in the IR, whereΣ has dimension n. After a Kaluza-Klein reduction of the D¼ 10=11 supergravity theory on M, to obtain a gravity theory in dþ 1 spacetime dimensions, these solutions may be viewed as black branes in AdSdþ1 with horizonΣ.
Such solutions were first constructed in the foundational work[1]and describe M5-branes and D3-branes wrapping Riemann surfaces of constant curvature, with genus g >1. Subsequently there have been many generalizations includ-ing wrappinclud-ing manifolds of higher dimension, relaxinclud-ing the constant curvature condition, allowing for punctures, etc. (e.g., Refs. [2–4]). In all these developments, super-symmetry is preserved by demanding that the field theory
arising on the brane world volume is “topologically twisted” [5,6]. This involves a specific coupling both to the metric onΣ and to external R-symmetry gauge fields. An important consequence of this twisting is that the Killing spinors preserved by the solution are independent of the coordinates ofΣ.
Here we discuss a class of supersymmetric solutions which have fundamentally new features. We present AdS3×Σ solutions of D ¼ 5 minimal gauged supergravity which we interpret as the near horizon limit of black brane solutions associated with D3-branes wrapped on Σ. The first new feature is that supersymmetry is not realized by a topological twist. The second is thatΣ is not a compact manifold but an orbifold [7]. More specifically, we will consider the weighted projective space Σ ¼ WCP1½n
−;nþ,
also known as a spindle. This is topologically a two-sphere but with conical deficit angles2πð1 − 1=n∓Þ at the poles, specified by two coprime positive integers n∓, with n− ≠ nþ [8].
Remarkably, after uplifting the AdS3×Σ solutions on specific Sasaki-Einstein five-manifolds, SE5, to obtain solutions of type IIB supergravity, they become completely smooth[9]. Moreover, the resulting AdS3×M7solutions are precisely those of Ref.[11]. Our construction suggests that the d¼ 2, N ¼ ð0; 2Þ SCFTs dual to the AdS3×M7 solutions of Ref. [11] arise from compactifying d¼ 4, N ¼ 1 SCFTs on a spindle, where the d ¼ 4 theories are dual to AdS5× SE5. This includes the case of N ¼ 4 SYM, where SE5¼ S5. We make a precision test of this interpretation: we compute the central charge and super-conformal R-symmetry of the d¼ 2 field theories using anomaly polynomials and c extremization [12] and find exact agreement with the gravity result[11].
D¼ 5 solutions.—The equations of motion for D ¼ 5 minimal gauged supergravity[13] are given by
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
Rμν¼ −4gμνþ 2 3FμρFνρ− 1 9gμνFρσFρσ; d F ¼ −2 3F∧ F; ð1Þ
where F¼ dA, with A the Abelian R-symmetry gauge field. A solution is supersymmetric if it admits a Killing spinor satisfying
∇μ−12i ðΓμνρ− 4δνμΓρÞFνρ−12Γμ− iAμ
ϵ ¼ 0; ð2Þ where ϵ is a Dirac spinor and fΓμ;Γνg ¼ 2gμν.
The supersymmetric solution of interest is given by ds25¼ 4y 9 ds2AdS3þ ds2Σ; A¼ 14 1 −a y dz: ð3Þ Here ds2AdS
3 is a unit radius metric on AdS3, while
ds2Σ¼ y qðyÞdy2þ
qðyÞ
36y2dz2; ð4Þ
is the metric on the horizonΣ and
qðyÞ ¼ 4y3− 9y2þ 6ay − a2; ð5Þ with a a constant.
Assuming a∈ ð0; 1Þ the three roots yiof qðyÞ are all real and positive. Defining y1< y2< y3, we then take y∈ ½y1; y2 to obtain a positive definite metric (4) on Σ. However, as y approaches y1 and y2 it is not possible to remove the conical deficit singularities at both roots by a single choice of period Δz for z, to obtain a smooth two-sphere. Instead we find that if
a¼ðn−− nþÞ 2ð2n −þ nþÞ2ðn−þ 2nþÞ2 4ðn2 −þ n−nþþ n2þÞ3 ; Δz ¼2ðn2−þ n−nþþ n2þÞ 3n−nþðn−þ nþÞ 2π; ð6Þ
then ds2Σis a smooth metric on the orbifoldΣ¼WCP1½n
−;nþ.
Specifically, there are conical deficit angles2πð1 − 1=n∓Þ at y¼ y1, y2, respectively, where n are arbitrary coprime
positive integers with n−> nþ.
Note that there is magnetic flux throughΣ: 1 2π Z ΣF¼ n−− nþ 2n−nþ : ð7Þ
This may be contrasted with the Euler number χðΣÞ ¼ 1 4π Z ΣRΣvolΣ¼ n−þ nþ n−nþ ; ð8Þ
where RΣ is the Ricci scalar of Σ, and volΣ is its volume form[14].
To solve Eq.(2)we writeΓa¼ γa⊗ σ3, for a¼ 0, 1, 2
with γ0¼ −iσ2, γ1¼ σ1, γ2¼ σ3, and Γ3¼ 1 ⊗ σ2, Γ4¼ 1 ⊗ σ1, where σi are Pauli matrices. We then write
ϵ ¼ ϑ ⊗ χ with ϑ a Killing spinor for AdS3 satisfying
∇aϑ ¼12γaϑ. The two-component spinor χ on the spindle is
given by χ ¼ ffiffiffiffiffiffiffiffiffiffiffi q1ðyÞ p ffiffiffi y p ; i ffiffiffiffiffiffiffiffiffiffiffi q2ðyÞ p ffiffiffi y p ; ð9Þ where
q1ðyÞ ¼ −a þ 2y3=2þ 3y; q2ðyÞ ¼ a þ 2y3=2− 3y; ð10Þ which satisfy qðyÞ ¼ q1ðyÞq2ðyÞ. In contrast to the topo-logical twist, this spinor depends on the coordinates ofΣ. Moreover, as shown in Ref. [15], the spinor is in fact a section of a nontrivial bundle overΣ. Note that the gauge choice used in Eq.(3)has been fixed by requiringϵ to be independent of z.
Uplift to IIB string theory.—Any supersymmetric solution to Eq.(1) uplifts (locally) to type IIB supergravity via[16]:
ds210¼ L2 ds25þ 1 3dψ þ σ þ 2 3A 2 þ ds2 KE4 gsF5¼ L4 4vol5−235F∧ J þ 2J ∧ J −2 3F∧ J ∧ 1 3dψ þ σ þ 2 3A : ð11Þ Here F5 is the self-dual five-form, gs is the string coupling constant, and L >0 is a length scale that is fixed by flux quantization. KE4 is an arbitrary positively curved Kähler-Einstein four-manifold with Kähler form J, normalized so that the Ricci form isR ¼ 6J, and σ is a local one-form with dσ ¼ 2J.
Substituting Eq.(3)into Eq.(11)we find that the D¼ 10 metric may be written as
ds210¼ 4
9L2y½ds2AdS3þ ds2M7; ð12Þ
whereM7 is a compact seven-manifold. This is the same solution of type IIB supergravity given in Ref.[11].
It was shown in Ref. [11] that M7 is the total space of a lens space S3=Zq fibration over the KE4, where the
twisting is parametrized by another positive integer p. The lens space fiber has coordinates y, z, ψ. In terms of our parameters n we identify
p¼ knþ; q¼k
Iðn−− nþÞ; ð13Þ where p; q∈ N are coprime. The Fano index I is the largest positive integer for whichRSc1=I∈ Z, for all two-cycles S in the KE4, where c1¼ ½R=2π ∈ H2ðKE4;ZÞ. We have also defined
k¼ hcfðI; pÞ; ð14Þ
and identify ψ with period Δψ given by[17]
Δψ ¼2πI
k : ð15Þ
At a fixed point in the D¼ 5 spacetime, the internal five-dimensional metric ds2SE
5 ¼ ð
1
3dψ þ σÞ2þ ds2KE4 in
Eq.(11)is then a regular Sasaki-Einstein manifold, which is simply connected when k¼ 1.
To obtain a string theory background one must also quantize the five-form flux F5 through all five-cycles in M7. This was carried out in Ref. [11]. We define the
integers M¼ Z KE4 c1∧ c1¼ 1 4π2 Z KE4 R ∧ R; ð16Þ
and h¼ hcfðM=I2; qÞ. Then if we choose L to satisfy L4 gsl4s ¼ 108π I3h k 2n þn−n; ð17Þ
wherelsis the string length and n∈ N, then one finds that
1=ð2πlsÞ4
R
DF5∈ Z, for all five-cycles D in M7.
There is a finite set of choices for the positively curved KE4. If KE4¼ CP2 then I¼ 3, M ¼ 9. For this case, k¼ 1 gives SE5¼ S5 as the internal space while k¼ 3 gives S5=Z3. If KE4¼ S2× S2 we have I¼ 2, M ¼ 8. Now k¼ 1 gives SE5¼ T1;1, while k¼ 2 gives SE5¼ T1;1=Z2. Finally, for KE4¼ dPm, 3 ≤ m ≤ 8, where dPm is a del Pezzo surface, we have I¼ 1, M¼ 9 − m.
A key observation is thatM7in the AdS3solutions of Ref. [11] may also be viewed as SE5 fibrations over Σ ¼ WCP1
½n−;nþ. We can begin with any weighted
projec-tive space, with weights n−> nþ, and then define p; q∈ Z via (13), where we also define
k¼ I
hcfðI; n−− nþÞ: ð18Þ
With this definition, p and q are manifestly coprime, and one can check that Eq.(14)is equivalent to Eq.(18). With this perspective, we can calculate the flux of F5through the SE5 fiber: N≡ 1 ð2πlsÞ4 Z SE5 F5¼ M I2hknþn−n∈ N: ð19Þ Notice that for a given spindle, specified by n, and a given
choice of KE4we only get a smooth type IIB solution for k as in Eq.(18)and hence a specific SE5. E.g., if KE4¼ CP2 and nþ¼ 2, then for n−¼ 3; 7; … and n− ¼ 5; 9; … we can uplift on S5=Z3 and S5, respectively.
The central charge is given by c¼ 3L=2Gð3Þ, where Gð3Þ is the Newton constant obtained by compactifying type IIB supergravity on M7 [18]. We can rewrite the result of Ref.[11] as c¼ 4ðn−− nþÞ 3 3n−nþðn2−þ n−nþþ n2þÞ a4d; ð20Þ where a4d≡ π 2N2 4volðSE5Þ : ð21Þ
In Ref.[11]the dual d¼ 2, N ¼ ð0; 2Þ SCFTs were not identified, but our D¼ 5 construction of the solutions, together with the flux condition(19), leads to a conjecture. Begin with the d¼ 4 SCFT dual to AdS5× SE5, describ-ing N D3-branes at the Calabi-Yau threefold sdescrib-ingularity with conical metric dr2þ r2ds2SE
5. The large N a-central
charge of this theory is precisely given by a4d in Eq.(21) [19]. One then compactifies that theory onΣ ¼ WCP1½n
−;nþ,
with a background R-symmetry gauge field with magnetic flux (7). The solutions we have described suggest the theory flows to a d¼ 2, N ¼ ð0; 2Þ SCFT in the IR, and we will give evidence for this below by computing the central charge via a field theory calculation.
The Uð1ÞR symmetry of the (0,2) theory dual to the AdS3×M7solutions is realized by a Killing vector, R2d, onM7. Using the results of Refs.[20,21]we deduce
R2d ¼ 2∂ψ þ 3n−nþðn−þ nþÞ
n2−þ n−nþþ n2þ∂φ: ð22Þ Here we have defined φ ¼2πzΔz so that Δφ ¼ 2π, and ∂φ generates the Uð1Þ isometry of the weighted projective spaceΣ, which we shall refer to as Uð1ÞJ. Note that the Killing spinor on the SE5has unit charge under R4d ¼ 2∂ψ, which may therefore be identified with the superconformal Uð1ÞRsymmetry of the d¼ 4 SCFT before compactifica-tion on Σ. In other words, Eq.(22) states that the d¼ 4 R-symmetry mixes with Uð1ÞJ in flowing to the d¼ 2 R-symmetry in the IR. We shall also recover Eq.(22)from a field theory calculation in the next section.
d¼ 4 SCFTs on Σ.—We begin with a general d ¼ 4 SCFT with anomaly polynomial given by the 6-form
A4d¼trR 3
6 c1ðR4dÞ3− trR
24c1ðR4dÞp1ðTZ6Þ: ð23Þ As is standard, in (even) dimension d the anomaly polynomial is a (dþ form on an abstract (d þ 2)-dimensional space, here called Z6. c1ðR4dÞ denotes the first Chern class of the d¼ 4 superconformal Uð1ÞR symmetry bundle over Z6, and p1 denotes the first Pontryagin class. The trace is over Weyl fermions when the theory has a Lagrangian description, and in any case we may always write
tr R3¼ 16
9 ð5a4d− 3c4dÞ; tr R¼ 16ða4d− c4dÞ; ð24Þ in terms of the central charges a4d, c4d. We focus on the large N limit in which a4d ¼ c4d to leading order, and hence
A4d ¼ 1627a4dc1ðR4dÞ3 ðat large NÞ: ð25Þ
We now compactify the d¼ 4 theory on Σ ¼ WCP1½n
−;nþ,
with magnetic flux (7) for the d¼ 4 R-symmetry gauge field. The resulting d¼ 2 anomaly polynomial will then capture the right-moving central charge cr, but crucially we
need to include the Uð1ÞJ global symmetry in d¼ 2 that comes from the isometry ofΣ. Geometrically, this involves taking Z6to be the total space of aΣ fibration over a four-manifold Z4[22]. More precisely, we let J be a Uð1Þ bundle over Z4, with connection corresponding to a background gauge field AJ for the d¼ 2 Uð1ÞJ global symmetry, and then fiber Σ over Z4 using the Uð1ÞJ action and connection AJ. In practice, this amounts to the replacement
dφ ↦ dφ þ AJ.
Incorporating the magnetic flux(7)into this construction amounts to“gauging” the Uð1Þ gauge field A in Eq.(3), as just described. Thus, we define the following connection one-form on Z6: A ¼1 4 1 −a y Δz 2πðdφ þ AJÞ ≡ ρðyÞðdφ þ AJÞ; ð26Þ
where recall that the gauge choice we made is such that the Killing spinors are uncharged under the Uð1ÞJ symmetry generated by∂φ. This is necessary for the twisting to make sense.A defined by Eq.(26)is a gauge field on Z6, which restricts to the supergravity gauge field A in Eq.(3)on each Σ fiber. We compute the curvature
F ¼ dA ¼ ρ0ðyÞdy ∧ ðdφ þ A
JÞ þ ρðyÞFJ; ð27Þ
where FJ¼ dAJ. The one-form dφ þ AJ is precisely the
global angular form for the Uð1Þ bundle, and so is globally defined on Z6 away from the poles ofΣ ¼ WCP1½n
−;nþ at
y¼ y1; y2. Moreover, one can verify that ρ0ðyÞdy ∧ dφ vanishes smoothly at the poles, where the angular coor-dinate φ is not defined, implying that F is a globally defined closed two-form on Z6. By construction, the integral of F over a fiber Σ of Z6 satisfies Eq. (7). More generally, the integrals of wedge products over the fibers are given, for s∈ N, by
Z Σ F 2π s ¼ 1 2s 1 ns þ− 1 ns − −FJ 2π s−1 : ð28Þ
The curvature formF defines a Uð1Þ bundle L over Z6 by taking c1ðLÞ ¼ ½F=2π ∈ H2ðZ6;RÞ. This is different from Ref.[22], where the Uð1Þ bundle was taken to be the tangent bundle to the fibers TfibersZ6, which gives the Euler
class(8), rather than Eq.(7). We note that at the poles we have c1ðLÞjy¼y1;y2 ¼ −2n1c1ðJÞ, where we have defined c1ðJÞ ¼ ½FJ=2π ∈ H2ðZ4;ZÞ. In the anomaly polynomial
we then write
c1ðR4dÞ ¼ c1ðR2dÞ þ c1ðLÞ; ð29Þ where R2dis the pullback of a Uð1Þ bundle over Z4. Notice that the twisting(29)will make sense globally only if the d¼ 4 R charges of fields satisfy appropriate quantization conditions, and for gauge-invariant operators this is equiv-alent to the global regularity and flux quantization con-ditions imposed on the supergravity solutions, cf. the discussion below Eq.(19).
The d¼ 2 anomaly polynomial is obtained by integrat-ingA4d in Eq. (25)over Σ. Using Eqs.(28)and(29) we compute A2d¼ 2 a4d 27 12 1 nþ− 1 n− c1ðR2dÞ2 −6 1 n2þ− 1 n2− c1ðR2dÞc1ðJÞþ 1 n3þ− 1 n3− c1ðJÞ2 : ð30Þ The coefficient of12c1ðLiÞc1ðLjÞ in A2dis trγ3QiQj, where the global symmetry Qiis associated to the Uð1Þ bundle Li
over Z4, andγ3is the d¼ 2 chirality operator. On the other hand, c extremization [12] implies that the d¼ 2 super-conformal Uð1ÞR extremizes
ctrial¼ 3tr γ3R2trial; ð31Þ over the space of possible R-symmetries. We set
Rtrial¼ R2dþ εJ; ð32Þ
and extremize the quadratic function ofε one obtains from Eqs.(30) and(31). The extremal value is
ε¼ 3n−nþðn−þ nþÞ
n2−þ n−nþþ n2þ: ð33Þ The right-moving central charge is given by Eq. (31)
evaluated on the superconformal R-symmetry [12]. Substituting Eq. (33)into Eqs. (31), (32)we find
cr¼
ðn−− nþÞ3
n−nþðn2−þ n−nþþ n2þÞ
4a4d
3 : ð34Þ
At leading order in N, cr¼ cl≡ c is the central charge of the SCFT, and we see that the field theory result (34)
precisely matches the gravity result (20). Moreover, the R-symmetry (32), with ε ¼ ε, precisely matches the
supergravity R-symmetry(22).
Discussion.—Our solutions exhibit a number of new properties, raising several directions for future research. First, despite having orbifold singularities in D¼ 5, when uplifted to D¼ 10 the solutions are completely regular. What type of singularities are permitted in lower-dimensional supergravity theories that have this property? Second, it is often claimed that supersymmetry requires a topological twist when branes wrap a compact manifold, so that the“twisted spinors” are constant on the manifold. Our near horizon solutions are a counterexample and it would be interesting to understand this more systematically. Third, our results suggest there should exist black string solutions which approach AdS5 in the UV and AdS3×Σ in the IR. Such solutions will reveal the precise deformations of the d¼ 4, N ¼ 1 SCFTs that can then flow to the d¼ 2, N ¼ ð0; 2Þ SCFTs dual to the AdS3×M7solutions of Ref. [11], including the way in which the D3-brane wrapping the spindle is preserving supersymmetry.
In Ref.[15]we present analogous supergravity solutions in D¼ 4, associated with M2-branes wrapped on a “spinning spindle.” In that case the full black hole solution is known and it approaches AdS4in the UV and AdS2×Σ in the IR, with a spindle horizonΣ. The D ¼ 4 black hole is accelerating and this leads to the conical deficit singular-ities onΣ. Once lifted to D ¼ 11 these orbifold singularities are removed and the solutions become completely regular. Supersymmetry is again not realized by the topological twist for the AdS2×Σ solution, similar to our AdS3×Σ solutions. In the UV, the black holes at finite temperature have a conformal boundary consisting of a spindle. In the supersymmetric and extremal limit, however, the spindle degenerates into“two halves,” each of which is associated with a topological twist, but with a different constant spinor on each component. It is another fascinating open question to determine how typical such a novel realization of supersymmetry is for branes wrapping spindles, as well as spaces of higher dimension.
Our results also suggest many questions on the field theory side. When defining a SCFT on a spindle what additional data should be specified at the orbifold points?
Considering weakly coupledN ¼ 4 SYM theory would be an important first step. Our holographic analysis shows that SCFTs dual to regular SE manifolds can only be placed on certain spindles and there is an apparent obstruction for those dual to irregular SE manifolds; why is this? The anomaly polynomial technique we employed is based on smooth manifolds and yet it gives a consistent result in the context of the orbifolds we studied, in the large N limit. It would be interesting to justify this more systematically (also see Ref.[23]) and determine subleading contributions. We thank K. Hristov and J. Lucietti for comments. This work was supported in part by STFC Grants No. ST/ P000762/1, No. ST/T000791/1 and No. ST/T000864/1. J. P. G. is supported as a KIAS Scholar and as a Visiting Fellow at the Perimeter Institute.
[1] J. M. Maldacena and C. Nunez,Int. J. Mod. Phys. A 16, 822 (2001).
[2] J. P. Gauntlett, N. Kim, and D. Waldram,Phys. Rev. D 63, 126001 (2001).
[3] D. Gaiotto and J. Maldacena, J. High Energy Phys. 10 (2012) 189.
[4] M. T. Anderson, C. Beem, N. Bobev, and L. Rastelli,
Commun. Math. Phys. 318, 429 (2013).
[5] E. Witten,Commun. Math. Phys. 117, 353 (1988). [6] M. Bershadsky, C. Vafa, and V. Sadov,Nucl. Phys. B463,
420 (1996).
[7] An orbifold is locally modelled on open subsets ofRn=Γ, where Γ are finite groups.
[8] WCP1½n−;nþ is a bad orbifold in the sense that it is not possible to move to a covering space that is a manifold. It does not admit a metric of constant curvature.
[9] The local D¼ 5 solutions were independently presented in Ref. [10], who also made a local connection with the solutions of Ref.[11].
[10] H. K. Kunduri, J. Lucietti, and H. S. Reall,J. High Energy Phys. 02 (2007) 026.
[11] J. P. Gauntlett, O. A. P. Mac Conamhna, T. Mateos, and D. Waldram,Phys. Rev. Lett. 97, 171601 (2006).
[12] F. Benini and N. Bobev,Phys. Rev. Lett. 110, 061601 (2013). [13] M. Gunaydin, G. Sierra, and P. Townsend, Nucl. Phys.
B242, 244 (1984).
[14] In general the integral of the curvature for a Uð1Þ gauge field on WCP1½n−;nþ is necessarily 2π=ðn−nþÞ times an integer e.g.,[15]. This makes the gauge field A, with flux given by
(7), a spincgauge field: n
−− nþis even/odd precisely when n−þ nþis even/odd, so that spinor fields with unit charge under A are always globally well-defined.
[15] P. Ferrero, J. P. Gauntlett, J. M. P. Ipiña, D. Martelli, and J. Sparks,arXiv:2012.08530.
[16] A. Buchel and J. T. Liu,Nucl. Phys. B771, 93 (2007). [17] The torus parametrised byðz; ψÞ is the same as [11]. We
have made identifications on z andψ in the opposite order to
[11], so(15)is not the same butΔψΔz is.
[18] J. Brown and M. Henneaux,Commun. Math. Phys. 104, 207 (1986).
[19] S. S. Gubser,Phys. Rev. D 59, 025006 (1999). [20] N. Kim,J. High Energy Phys. 01 (2006) 094.
[21] J. P. Gauntlett, N. Kim, and D. Waldram,J. High Energy Phys. 04 (2007) 005.
[22] S. M. Hosseini, K. Hristov, Y. Tachikawa, and A. Zaffaroni,
J. High Energy Phys. 09 (2020) 167.
[23] I. Bah, F. Bonetti, R. Minasian, and P. Weck,J. High Energy Phys. 02 (2021) 116.
